# What are some good introductory books on mathematical proofs?

There was a time when I avoided math proofs, but now I am starting to enjoy them. I am taking Intro to Linear Algebra and am falling in love with proofs. Are there any introduction to mathematical proofs books that blow the others out of the water?

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If you want a historical view of great theorems (and their proofs), you should check out Journey Through Genius. I love that book. It's honestly what made me fall in love with mathematics, especially the two chapters on set theory and Cantor's proofs and arguments. I read the whole book my senior year in high school; most of it was fairly accessible but I struggled a little with the diagonalization argument by Cantor. I think it took me four or five read throughs before I began understanding it. – Cameron Williams Jan 29 '14 at 3:20
I thoroughly recommend George Polya, How to Solve It. A great collection of ideas for thinking about problems. Just don't fall into the trap (as I did when I was very young) of imagining that it's a complete solution manual for everything ;-) – David Jan 29 '14 at 3:24
I don't know about "blowing other books out of the water", but one I'm looking at just now is the recent Charming Proofs: A Journey Into Elegant Mathematics, by Alsina and Nelsen. There is also Nelsen's Proofs without Words , which is good for emphasizing ways to think about the relationships described in a proposition, which is important in developing the "imaginative" part of mathematical reasoning. (Obviously, you couldn't just turn in a diagram as a proof...) Does anyone know how Aigner and Zeigler's Proofs from THE BOOK is? (Google 'Erdos' if you don't know what "The Book" is.) – RecklessReckoner Jan 29 '14 at 3:45
@RecklessReckoner For a second I thought you were keeping the title of that book a secret. I skimmed through it and it looks amazing! I will start reading it ASAP! Thanks! – zerosofthezeta Jan 29 '14 at 4:36
Do you mean "The Book"? No, it's not like Smullyan's What Is The Name of This Book? ... – RecklessReckoner Jan 29 '14 at 4:39

## 2 Answers

George Polya's How to Solve It immediately comes to mind. I know many now fantastic pre-mathematicians who learned calculus and the basics of analysis from Spivak's Calculus and even if you know the material to go back and do it again in a formal way is very healthy. In addition Proofs from THE BOOK was mentioned above and was recommended to me by Ngo Bao Chao when I asked about books to study problem-solving techniques from. I don't mean to come off as name-dropping but I feel that (as he is a fields medalist) his advice is worth heeding. I, personally, really liked it.

However I have to make note that I think if you'd phrased your question as "should I read a book about proofs to learn proofs" my response would be an emphatic no. In my experience if you don't see proofs by doing some fun mathematics you will not get much better about doing them yourself. Just reading about how to prove things can only get you so far before you're sort of stumped as to how to proceed. I would say the better approach is to find a rigorous treatment of a subject that you're very interested in, and read that, following along with the proofs of the theorems in the book and eventually trying to do them yourself without looking at the proofs given.

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I quite agree with you that merely reading about proofs is not in itself a good way to learn to do your own proofs. ("Math is not a spectator sport," as we often tell students in courses.) I think, though, that just as reading "good" literature is helpful in learning how to write better in your language, studying well-constructed proofs (and also seeing the variety of ways in which a proposition may sometimes be demonstrated) can be helpful in learning "good" argumentative style. (Geometry was part of the classical curriculum in part because it taught lawyers how to form strong arguments.) – RecklessReckoner Jan 29 '14 at 4:07

One option is to read an introductory book on a topic that interests you. For example, if you are interested in number theory, you can read Harold Stark's An Introduction to Number Theory. Depending on your motivation and degree of comfort reading proofs at this level, something like this might be a good option - an "introduction to proofs" book isn't a necessity for everyone.

However, if you want a book that is geared specifically for those who are just starting out with rigorous math and are still getting used to proofs, you might enjoy Journey into Mathematics: An Introduction to Proofs by Joseph Rotman. Unlike some such books, it doesn't dwell on trivialities about logic and sets. Instead, it discusses interesting yet accessible topics in elementary mathematics like Pythagorean triples, the number $\pi$, and cubic and quartic equations. Along the way, it introduces important concepts such as proof by induction, the formal definition of convergence of a sequence, and complex numbers. The book makes use of calculus, taking advantage of the fact that most North American students at this "transition to advanced mathematics" stage have already had courses in calculus. But what you will remember after reading it ought to be the actual mathematics in it, so you hopefully won't feel as if you've wasted your time.

It's not that sets and logic can't be interesting in themselves, but usually the more interesting aspects of these subjects can only be appreciated once a learner is well acquainted with mathematical methods in general.

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